Best way to denote some trigonometric functions ("tg" vs "tan", "ctg" vs "cot")

In current US textbooks, $\tan$ and $\cot$ are commonly used and standardized. Also: $\sin, \cos, \sec, \csc$. In other countries, and in the 19th century, you will find others.

Standard

ISO 80000-2 Quantities and units — Part 2: Mathematical signs and symbols to be used in the natural sciences and technology

is perfectly clear: the right symbols are $\tan x$ (item 2-13.4) and $\cot x$ (item 2-13.5).

To cite the standard: "$\text{tg } x$, $\text{ctg } x$ should not be used."

Some people are familiar with tg and some people are familiar with tan.So, for wide use just define the notation.It solve your's problem I think.